MR. A. CAYLEY’S SIXTH MEMOIR UPON QUANTICS. 
85 
of the distances of points upon different lines, or of lines through different points, but 
of the distances of points on a line and of lines through a point. The pole of any line 
in relation to the Absolute may be termed simply the pole, and in like manner the polar 
of any line in relation to the Absolute may be termed simply the polar, and we have 
the theorem that the distance of two points or lines is equal to the distance of their 
polars or poles, or what is the same thing, that the distance of two poles and the distance 
of the two corresponding polars are equal. And we may, as a definition, establish the 
notion of the distance of a point from a line, viz. it is the complement of the distance 
of the polar of the point from the line, or what is the same thing, the complement of 
the distance of the point from the pole of the line. The distance of a pole and polar is 
therefore the complement of zero, that is, it is the quadrant. 
215. It has, by means of the preceding assumption as to the quadrant, been possible 
to establish the notion of distance, without the assistance of the circle, but this figure 
must now be considered. A conic inscribed in the Absolute is termed a circle ; the 
centre of inscription (or point of intersection of the common tangents) and the axis of 
inscription (or line of junction of the common ineunts) are the centre and axis of the 
circle. All the points of a circle are equidistant from the centre ; all the tangents are 
equidistant from the axis, and this distance is the complement of the former distance. 
216. These properties of the circle lead immediately to the analytical expressions for 
the distances of points or lines in terms of the coordinates. In fact, take 
{a, b, c,f, g, hjx, y, zf={) 
for the point-equation of the Absolute ; its line-equation will be 
(^, 33, C, JT, (§, III, 
The point-equation of the circle having the point {cc\ y, z') for its centre, is 
(a, ..J^, y, z)\a, ..Jx\ y\ cos" ^— {(«, y, zjx', y\ z’)y=-0, 
or 
(a, ..Jx, y, zy(a, y', 2 ')" sin" ^—(91, ..'Jyyz! —y'%, zx'—z'x, xy'~x'yy=0, 
from which (by the same reasoning as for the case of geometry of one dimension) it^ 
follows that the distance of the points {x, y, 2 ), {cd, y\ z') is 
cos"' {a, 
s/ (a, ..Xx, y, zy^y {a, ..Xx', y', 2 ')" 
or what is the same thing, 
1 \/ ..Xy^'—]f^, zod—z'x, xy’-^yf . 
\/ {a, ..Xx, y, zy\/ (a, . . y', dy 
and it appears from the cosine formula (see ante., No. 208), that if P, P', P ' be points 
on the same line, then we have, as we ought to have, 
Dist. (P, P')+ Dist. (F, P")= Dist. (P, P"). 
217. In like manner, the line-equation of the same circle, the line-coordinates of the 
MDCCCLIX. s 
